prove $H(x)=x^\top x$ is constant along solutions of the sytem if $A(x)^\top + A(x)=0$

Question

Could someone please help me to understand the following:

Having the differential equation: $\dot{x} = A(x)x$ where $A(x)$ is a real -valued matrix of dimension $n\times n$

How can I prove that the function $H(x)=x^\top x$ is constant along solutions of the sytem if $A(x)^\top + A(x)=0$

And how to prove that the origin is a Lyapunov stable fixed point if $A(x)^\top + A(x) \prec 0$

Thanks!

Answer 1

When in doubt, differentiate! $$ \frac{d}{dt} H(x(t)) = \dot{x}^Tx + x^T\dot{x} = (A(x)x)^T x + x^TA(x)x = x^T(A^T(x) + A(x))x = 0. $$

Now for stability, observe that when $A^T(x) + A(x) \prec 0$, the right hand side of the above equation is negative, so $\frac{d}{dt}H(x(t)) < 0$. Thus, the norm-squared $x^Tx$ of the solution will be decreasing as a function of time.

Hope that helps.